Angular Kinematics
Kinematics • The form, pattern, or sequencing of movement with respect to time • Forces causing the motion are not considered Angular Motion (Rotation) • All points in an object or system move in a circle about a single axis of rotation. All points move through the same angle in the same time Angular Kinematics • The kinematics of particles, objects, or systems undergoing angular motion

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Angular Kinematics & Motion
• Most volitional movement is performed through rotation of the body segments • The body is often analyzed as a collection of rigid, rotating segments linked at the joint centers • This is a rough approximation
SHOULDER NECK

Example Problem #1
A figure skater spins 10.5 revolutions in a clockwise direction, pauses, then spins 60° counterclockwise before skating away. What were the total angular distance and angular displacement?

Computing Joint Angles
• Involves subtracting absolute angles of segments • Exact formula and order of subtraction depends on the joint and the convention chosen

θknee = θleg– θthigh
HIP

θknee = 180° + θthigh– θleg
HIP

θthigh θleg
KNEE ANKLE

θthigh θknee
KNEE

θleg

θknee

ANKLE

Angular Velocity
• The rate of change in the absolute or relative angular position or orientation of a line segment
change in angular angular position = = velocity change in time angular displacement change in time

• Shorthand notation:

ω=

θfinal – θinitial tfinal – tinitial

=

∆θ ∆t

• Has units of (angular units)/time (e.g. radians/s, °/s)

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Angular Speed
• The angular distance traveled divided by the time taken to cover it • Equal to the average magnitude of the instantaneous angular velocity over that time.

Example Problem #2
The figure skater of Problem #1 completes the first (clockwise) spin in 3 s, pauses for 1 s, then completes the second (counterclockwise) spin in 0.3 s. What were her average angular velocity and average angular speed during the first spin? What were her average angular velocity and average angular speed for the skill as a whole?

Example Problem #3
A person is performing a squat exercise. She starts from a standing (i.e. anatomical) position. At her lowest point, 2 seconds later, her knees are flexed to 60° and her hips are flexed to 90° from the anatomical position. 1 second later, she has risen back to the standing position and completed the exercise. What were the average knee and hip angular velocities during each phase of the exercise? for the exercise as a whole?

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Average vs. Instantaneous Velocity
• Previous formulas give us the average velocity between an initial time (t1) and a final time (t2) • Instantaneous angular velocity is the angular velocity at a single instant in time • Can estimate instantaneous angular velocity using the central difference method:

ω (at t1) =

θ (at t1 + ∆t) – θ (at t1 – ∆t) 2 ∆t

where ∆t is a very small change in time

Angular Velocity as a Slope
• Graph of angular position vs. time slope = instantaneous ω at t1 ∆θ(1→2) ∆t t1 ∆t(1→2) t2 time (s) slope = average ω from t1 to t2